12n
0417
(K12n
0417
)
A knot diagram
1
Linearized knot diagam
3 6 8 11 2 10 3 7 1 4 10 7
Solving Sequence
3,7
8
4,10
11 5 6 2 1 9 12
c
7
c
3
c
10
c
4
c
6
c
2
c
1
c
9
c
12
c
5
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h536u
11
− 335u
10
+ ··· + 239b + 523, −947u
11
+ 562u
10
+ ··· + 239a − 1324,
u
12
− 4u
10
+ 13u
8
− 2u
7
− 24u
6
− 3u
5
+ 18u
4
+ 3u
3
− 7u
2
− u + 1i
I
u
2
= hu
6
− 3u
4
+ 3u
2
+ b + u − 1, −u
6
+ u
5
+ 4u
4
− 3u
3
− 5u
2
+ a + 2u + 3, u
7
− 4u
5
+ 6u
3
+ u
2
− 4u − 1i
I
u
3
= h68215362482207u
19
+ 127486835274380u
18
+ ··· + 1057281252711b + 705066157444833,
298873575023330u
19
+ 558685165549478u
18
+ ··· + 3171843758133a + 3089653339595532,
u
20
+ u
19
+ ··· + 18u − 9i
I
u
4
= hu
6
− 3u
4
− u
3
+ 2u
2
+ b − 2, −u
7
+ 3u
5
+ u
4
− 2u
3
+ u
2
+ a + 2u − 2, u
8
− 4u
6
− u
5
+ 5u
4
+ u
3
− 4u
2
+ 1i
* 4 irreducible components of dim
C
= 0, with total 47 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= h536u
11
− 335u
10
+ · · · + 239b + 523, −947u
11
+ 562u
10
+ · · · +
239a − 1324, u
12
− 4u
10
+ · · · − u + 1i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
−u
−u
3
+ u
a
10
=
3.96234u
11
− 2.35146u
10
+ ··· − 11.8033u + 5.53975
−2.24268u
11
+ 1.40167u
10
+ ··· + 5.48954u − 2.18828
a
11
=
3.96234u
11
− 2.35146u
10
+ ··· − 11.8033u + 5.53975
−2.24268u
11
+ 1.40167u
10
+ ··· + 5.48954u − 2.18828
a
5
=
2.35146u
11
− 1.71967u
10
+ ··· − 10.5021u + 3.96234
−1.40167u
11
+ 1.25105u
10
+ ··· + 5.43096u − 2.24268
a
6
=
0.589958u
11
− 0.493724u
10
+ ··· + 1.58577u + 0.543933
2.71130u
11
− 1.69456u
10
+ ··· − 8.15900u + 3.13808
a
2
=
3.23431u
11
− 2.14644u
10
+ ··· − 7.33473u + 3.97490
0.974895u
11
− 0.234310u
10
+ ··· − 3.53556u + 1.35983
a
1
=
3.23431u
11
− 2.14644u
10
+ ··· − 7.33473u + 3.97490
−0.991632u
11
+ 0.744770u
10
+ ··· + 1.84519u − 0.786611
a
9
=
−u
2
+ 1
u
2
a
12
=
2.24268u
11
− 1.40167u
10
+ ··· − 5.48954u + 3.18828
−0.991632u
11
+ 0.744770u
10
+ ··· + 1.84519u − 0.786611
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1675
239
u
11
+
1734
239
u
10
+
5437
239
u
9
−
5818
239
u
8
−
17601
239
u
7
+
22148
239
u
6
+
23370
239
u
5
−
22069
239
u
4
−
17044
239
u
3
+
13395
239
u
2
+
5056
239
u −
6743
239
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 6u
11
+ ··· + 848u + 64
c
2
, c
5
u
12
+ 8u
11
+ ··· + 52u + 8
c
3
, c
4
, c
7
c
10
u
12
− 4u
10
+ 13u
8
+ 2u
7
− 24u
6
+ 3u
5
+ 18u
4
− 3u
3
− 7u
2
+ u + 1
c
6
, c
9
u
12
+ u
11
+ ··· + 8u + 1
c
8
, c
11
u
12
+ 8u
11
+ ··· + 15u + 1
c
12
u
12
+ 10u
11
+ ··· − 196u − 20
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 18y
11
+ ··· − 419072y + 4096
c
2
, c
5
y
12
− 6y
11
+ ··· − 848y + 64
c
3
, c
4
, c
7
c
10
y
12
− 8y
11
+ ··· − 15y + 1
c
6
, c
9
y
12
+ 11y
11
+ ··· − 16y + 1
c
8
, c
11
y
12
+ 20y
11
+ ··· − 43y + 1
c
12
y
12
− 40y
11
+ ··· − 4496y + 400
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.790520 + 0.392533I
a = −0.333077 − 0.431942I
b = 0.908183 + 0.435278I
−3.78101 + 0.45111I −18.5516 − 2.4488I
u = −0.790520 − 0.392533I
a = −0.333077 + 0.431942I
b = 0.908183 − 0.435278I
−3.78101 − 0.45111I −18.5516 + 2.4488I
u = −0.736086 + 0.101541I
a = 1.55509 + 1.56812I
b = 0.505873 − 0.967109I
−0.21900 + 4.19269I −13.9112 − 5.3421I
u = −0.736086 − 0.101541I
a = 1.55509 − 1.56812I
b = 0.505873 + 0.967109I
−0.21900 − 4.19269I −13.9112 + 5.3421I
u = 0.653112 + 0.249393I
a = 1.71916 − 0.21403I
b = −0.023047 + 0.696086I
−0.134851 + 0.667722I −12.59936 + 0.82514I
u = 0.653112 − 0.249393I
a = 1.71916 + 0.21403I
b = −0.023047 − 0.696086I
−0.134851 − 0.667722I −12.59936 − 0.82514I
u = 1.54915
a = −0.373370
b = 0.477327
−15.2923 −6.68440
u = 0.403757
a = 0.897573
b = 0.248749
−0.603932 −16.3720
u = −1.36080 + 0.93328I
a = −0.627613 − 0.831271I
b = −1.09936 + 1.61015I
5.78267 + 12.88920I −12.7241 − 6.3496I
u = −1.36080 − 0.93328I
a = −0.627613 + 0.831271I
b = −1.09936 − 1.61015I
5.78267 − 12.88920I −12.7241 + 6.3496I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.25783 + 1.10048I
a = −0.575658 + 0.547864I
b = −0.15469 − 1.93823I
7.12277 − 3.83904I −10.68570 + 2.08253I
u = 1.25783 − 1.10048I
a = −0.575658 − 0.547864I
b = −0.15469 + 1.93823I
7.12277 + 3.83904I −10.68570 − 2.08253I
6
II. I
u
2
= hu
6
− 3u
4
+ 3u
2
+ b + u − 1, −u
6
+ u
5
+ 4u
4
− 3u
3
− 5u
2
+ a + 2u +
3, u
7
− 4u
5
+ 6u
3
+ u
2
− 4u − 1i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
−u
−u
3
+ u
a
10
=
u
6
− u
5
− 4u
4
+ 3u
3
+ 5u
2
− 2u − 3
−u
6
+ 3u
4
− 3u
2
− u + 1
a
11
=
u
6
− u
5
− 4u
4
+ 3u
3
+ 6u
2
− 2u − 3
−u
6
+ 4u
4
− 4u
2
− u + 1
a
5
=
−u
6
+ 3u
4
− 3u
2
+ 1
u
3
− 2u − 1
a
6
=
−u
6
+ 4u
4
− u
3
− 6u
2
+ u + 4
−u
5
− u
4
+ 3u
3
+ 2u
2
− 3u − 2
a
2
=
u
6
− 4u
4
+ 5u
2
+ u − 3
u + 1
a
1
=
u
6
− 4u
4
+ 5u
2
+ u − 3
u
4
− u
2
+ 1
a
9
=
−u
2
+ 1
u
2
a
12
=
u
6
− 3u
4
+ 4u
2
+ u − 2
u
4
− u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
6
+ 4u
5
+ 13u
4
− 10u
3
− 11u
2
+ 4u − 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
− 7u
6
+ 21u
5
− 37u
4
+ 37u
3
− 24u
2
+ 9u − 1
c
2
u
7
+ u
6
− 3u
5
− 5u
4
+ u
3
+ 4u
2
+ u − 1
c
3
, c
10
u
7
− 4u
5
+ 6u
3
− u
2
− 4u + 1
c
4
, c
7
u
7
− 4u
5
+ 6u
3
+ u
2
− 4u − 1
c
5
u
7
− u
6
− 3u
5
+ 5u
4
+ u
3
− 4u
2
+ u + 1
c
6
, c
9
u
7
+ u
6
+ u
4
− u
3
− u
2
− u − 1
c
8
, c
11
u
7
+ 8u
6
+ 28u
5
+ 56u
4
+ 68u
3
+ 49u
2
+ 18u + 1
c
12
u
7
+ 7u
6
+ 17u
5
+ 30u
4
+ 45u
3
+ 27u
2
+ 6u + 4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
− 7y
6
− 3y
5
− 133y
4
− 43y
3
+ 16y
2
+ 33y −1
c
2
, c
5
y
7
− 7y
6
+ 21y
5
− 37y
4
+ 37y
3
− 24y
2
+ 9y −1
c
3
, c
4
, c
7
c
10
y
7
− 8y
6
+ 28y
5
− 56y
4
+ 68y
3
− 49y
2
+ 18y −1
c
6
, c
9
y
7
− y
6
− 4y
5
− y
4
+ 5y
3
+ 3y
2
− y −1
c
8
, c
11
y
7
− 8y
6
+ 24y
5
− 76y
4
+ 128y
3
− 65y
2
+ 226y −1
c
12
y
7
− 15y
6
− 41y
5
+ 264y
4
+ 553y
3
− 429y
2
− 180y −16
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.970575 + 0.467764I
a = 0.61390 + 1.62676I
b = 0.307176 − 1.007690I
−0.15907 + 5.95632I −10.30399 − 9.49474I
u = −0.970575 − 0.467764I
a = 0.61390 − 1.62676I
b = 0.307176 + 1.007690I
−0.15907 − 5.95632I −10.30399 + 9.49474I
u = 1.28252
a = −0.853969
b = −1.55070
−9.96367 −15.2580
u = 1.219310 + 0.473158I
a = 0.831288 − 0.514652I
b = −0.187678 + 0.823913I
−1.96867 − 1.64297I −12.60937 + 1.83263I
u = 1.219310 − 0.473158I
a = 0.831288 + 0.514652I
b = −0.187678 − 0.823913I
−1.96867 + 1.64297I −12.60937 − 1.83263I
u = −1.52200
a = 0.182606
b = −0.759603
−15.6521 −29.3740
u = −0.257994
a = −2.21901
b = 1.07131
−3.02746 −9.54090
10
III.
I
u
3
= h6.82 × 10
13
u
19
+ 1.27 × 10
14
u
18
+ · · · + 1.06 × 10
12
b + 7.05 × 10
14
, 2.99 ×
10
14
u
19
+5.59×10
14
u
18
+· · ·+3.17×10
12
a+3.09×10
15
, u
20
+u
19
+· · ·+18u−9i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
−u
−u
3
+ u
a
10
=
−94.2271u
19
− 176.139u
18
+ ··· + 828.728u − 974.087
−64.5196u
19
− 120.580u
18
+ ··· + 565.622u − 666.867
a
11
=
−27.9506u
19
− 52.2643u
18
+ ··· + 246.326u − 288.781
−80.6876u
19
− 150.746u
18
+ ··· + 707.746u − 833.790
a
5
=
4.98357u
19
+ 9.51307u
18
+ ··· − 45.9289u + 48.9194
−99.9143u
19
− 186.269u
18
+ ··· + 873.984u − 1037.72
a
6
=
−68.8452u
19
− 128.348u
18
+ ··· + 604.749u − 717.167
−76.2363u
19
− 142.302u
18
+ ··· + 674.421u − 794.311
a
2
=
69.0216u
19
+ 129.223u
18
+ ··· − 608.826u + 712.093
105.226u
19
+ 196.557u
18
+ ··· − 922.384u + 1090.86
a
1
=
69.0216u
19
+ 129.223u
18
+ ··· − 608.826u + 712.093
52.6964u
19
+ 98.1510u
18
+ ··· − 459.959u + 549.054
a
9
=
−u
2
+ 1
u
2
a
12
=
121.718u
19
+ 227.374u
18
+ ··· − 1068.79u + 1261.15
52.6964u
19
+ 98.1510u
18
+ ··· − 459.959u + 549.054
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
128749727823573
352427084237
u
19
−
240297448875642
352427084237
u
18
+ ··· +
1127484403652596
352427084237
u −
1338735790487557
352427084237
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
10
+ u
9
+ 8u
8
+ 12u
7
− 8u
6
− 6u
5
− 4u
4
+ 21u
3
+ 21u
2
+ 14u + 1)
2
c
2
, c
5
(u
10
− 3u
9
+ 4u
8
− 2u
7
+ 2u
6
− 4u
5
− u
3
+ 5u
2
− 2u − 1)
2
c
3
, c
4
, c
7
c
10
u
20
− u
19
+ ··· − 18u − 9
c
6
, c
9
u
20
− 3u
19
+ ··· − 6u − 1
c
8
, c
11
u
20
+ 5u
19
+ ··· + 576u + 81
c
12
(u
10
− 4u
9
− u
8
+ 23u
7
− 31u
6
+ 7u
5
+ 6u
4
+ 4u
3
− 10u
2
+ 5u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
10
+ 15y
9
+ ··· − 154y + 1)
2
c
2
, c
5
(y
10
− y
9
+ 8y
8
− 12y
7
− 8y
6
+ 6y
5
− 4y
4
− 21y
3
+ 21y
2
− 14y + 1)
2
c
3
, c
4
, c
7
c
10
y
20
− 5y
19
+ ··· − 576y + 81
c
6
, c
9
y
20
+ 9y
19
+ ··· + 144y + 1
c
8
, c
11
y
20
+ 7y
19
+ ··· − 23004y + 6561
c
12
(y
10
− 18y
9
+ ··· − 5y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.388171 + 0.947508I
a = 0.0090099 + 0.1229930I
b = −1.35420 + 0.42025I
2.99056 − 3.56450I −10.26354 + 3.14852I
u = 0.388171 − 0.947508I
a = 0.0090099 − 0.1229930I
b = −1.35420 − 0.42025I
2.99056 + 3.56450I −10.26354 − 3.14852I
u = −0.992473 + 0.348531I
a = 0.17968 + 1.68020I
b = 0.0378360 − 0.0943430I
−1.26454 + 5.52673I −18.1850 − 5.9531I
u = −0.992473 − 0.348531I
a = 0.17968 − 1.68020I
b = 0.0378360 + 0.0943430I
−1.26454 − 5.52673I −18.1850 + 5.9531I
u = 0.872812 + 0.011238I
a = 1.102060 − 0.425174I
b = 0.274355 − 0.471998I
−0.136787 − 0.639555I −12.53187 − 0.40948I
u = 0.872812 − 0.011238I
a = 1.102060 + 0.425174I
b = 0.274355 + 0.471998I
−0.136787 + 0.639555I −12.53187 + 0.40948I
u = 0.866881
a = −1.65529
b = −1.53301
−12.1512 −23.7630
u = 1.001390 + 0.640706I
a = 0.418488 − 1.217960I
b = 0.531023 + 1.236570I
−1.26454 − 5.52673I −18.1850 + 5.9531I
u = 1.001390 − 0.640706I
a = 0.418488 + 1.217960I
b = 0.531023 − 1.236570I
−1.26454 + 5.52673I −18.1850 − 5.9531I
u = 0.468108 + 0.513336I
a = 1.40798 − 1.10900I
b = −0.102915 + 0.895664I
−0.136787 + 0.639555I −12.53187 + 0.40948I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.468108 − 0.513336I
a = 1.40798 + 1.10900I
b = −0.102915 − 0.895664I
−0.136787 − 0.639555I −12.53187 − 0.40948I
u = −0.605174
a = 0.762515
b = 1.92560
−7.31763 −2.93760
u = −1.07994 + 0.93851I
a = 0.617818 + 0.827405I
b = 0.59658 − 1.52167I
2.99056 + 3.56450I −10.26354 − 3.14852I
u = −1.07994 − 0.93851I
a = 0.617818 − 0.827405I
b = 0.59658 + 1.52167I
2.99056 − 3.56450I −10.26354 + 3.14852I
u = −1.45778
a = −1.23121
b = −0.970112
−7.31763 −2.93760
u = 0.91975 + 1.23722I
a = −0.446520 + 0.963248I
b = −0.95499 − 1.76369I
8.14515 − 4.60681I −10.16934 + 2.47582I
u = 0.91975 − 1.23722I
a = −0.446520 − 0.963248I
b = −0.95499 + 1.76369I
8.14515 + 4.60681I −10.16934 − 2.47582I
u = −0.63701 + 1.40514I
a = −0.447242 − 0.786229I
b = 0.22554 + 1.94609I
8.14515 − 4.60681I −10.16934 + 2.47582I
u = −0.63701 − 1.40514I
a = −0.447242 + 0.786229I
b = 0.22554 − 1.94609I
8.14515 + 4.60681I −10.16934 − 2.47582I
u = −1.68554
a = −0.225221
b = −0.928948
−12.1512 −23.7630
15
IV. I
u
4
= hu
6
− 3u
4
− u
3
+ 2u
2
+ b − 2, −u
7
+ 3u
5
+ u
4
− 2u
3
+ u
2
+ a +
2u − 2, u
8
− 4u
6
− u
5
+ 5u
4
+ u
3
− 4u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
−u
−u
3
+ u
a
10
=
u
7
− 3u
5
− u
4
+ 2u
3
− u
2
− 2u + 2
−u
6
+ 3u
4
+ u
3
− 2u
2
+ 2
a
11
=
u
7
− 3u
5
− u
4
+ 2u
3
− u
2
− 2u + 1
−u
6
+ 3u
4
+ u
3
− 3u
2
+ 3
a
5
=
u
7
− u
6
− 4u
5
+ 2u
4
+ 6u
3
− u
2
− 4u + 2
3u
7
− 11u
5
− 3u
4
+ 11u
3
+ 2u
2
− 8u
a
6
=
−2u
7
+ 7u
5
+ 2u
4
− 6u
3
− u
2
+ 4u − 1
u
7
+ 2u
6
− 3u
5
− 8u
4
+ 6u
2
− u − 4
a
2
=
−u
7
+ u
6
+ 4u
5
− 3u
4
− 6u
3
+ 4u
2
+ 5u − 3
u
6
− 4u
4
− u
3
+ 5u
2
+ 2u − 4
a
1
=
−u
7
+ u
6
+ 4u
5
− 3u
4
− 6u
3
+ 4u
2
+ 5u − 3
u
6
− 4u
4
− u
3
+ 4u
2
+ u − 3
a
9
=
−u
2
+ 1
u
2
a
12
=
−u
7
+ 2u
6
+ 4u
5
− 7u
4
− 7u
3
+ 8u
2
+ 6u − 6
u
6
− 4u
4
− u
3
+ 4u
2
+ u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 15u
7
+ 5u
6
− 53u
5
− 35u
4
+ 49u
3
+ 28u
2
− 33u − 25
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
− 4u
3
+ 6u
2
− 5u + 1)
2
c
2
(u
4
− 2u
2
+ u + 1)
2
c
3
, c
10
u
8
− 4u
6
+ u
5
+ 5u
4
− u
3
− 4u
2
+ 1
c
4
, c
7
u
8
− 4u
6
− u
5
+ 5u
4
+ u
3
− 4u
2
+ 1
c
5
(u
4
− 2u
2
− u + 1)
2
c
6
, c
9
u
8
− 4u
7
+ 5u
6
− u
5
− 4u
4
+ 5u
3
− 2u
2
+ 1
c
8
, c
11
u
8
+ 8u
7
+ 26u
6
+ 49u
5
+ 61u
4
+ 49u
3
+ 26u
2
+ 8u + 1
c
12
(u
4
− 4u
3
+ 4u
2
− u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
− 4y
3
− 2y
2
− 13y + 1)
2
c
2
, c
5
(y
4
− 4y
3
+ 6y
2
− 5y + 1)
2
c
3
, c
4
, c
7
c
10
y
8
− 8y
7
+ 26y
6
− 49y
5
+ 61y
4
− 49y
3
+ 26y
2
− 8y + 1
c
6
, c
9
y
8
− 6y
7
+ 9y
6
− 5y
5
+ 8y
4
+ y
3
− 4y
2
− 4y + 1
c
8
, c
11
y
8
− 12y
7
+ 14y
6
+ 39y
5
+ 145y
4
+ 39y
3
+ 14y
2
− 12y + 1
c
12
(y
4
− 8y
3
+ 10y
2
+ 7y + 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744022 + 0.443105I
a = 1.42333 − 1.57383I
b = 0.758066 + 0.777626I
−0.24852 − 1.96274I −13.4834 + 4.4361I
u = 0.744022 − 0.443105I
a = 1.42333 + 1.57383I
b = 0.758066 − 0.777626I
−0.24852 + 1.96274I −13.4834 − 4.4361I
u = −0.992148 + 0.590877I
a = 1.145330 + 0.258001I
b = 0.241934 − 0.777626I
−0.24852 − 1.96274I −13.4834 + 4.4361I
u = −0.992148 − 0.590877I
a = 1.145330 − 0.258001I
b = 0.241934 + 0.777626I
−0.24852 + 1.96274I −13.4834 − 4.4361I
u = 0.701343
a = 0.127840
b = 1.96805
−7.58970 −33.0820
u = −1.42584
a = −1.41328
b = −0.968048
−7.58970 −33.0820
u = −0.561188
a = 2.50424
b = 1.45971
−11.6525 −6.95090
u = 1.78193
a = −0.356111
b = −0.459710
−11.6525 −6.95090
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
4
− 4u
3
+ 6u
2
− 5u + 1)
2
· (u
7
− 7u
6
+ 21u
5
− 37u
4
+ 37u
3
− 24u
2
+ 9u − 1)
· (u
10
+ u
9
+ 8u
8
+ 12u
7
− 8u
6
− 6u
5
− 4u
4
+ 21u
3
+ 21u
2
+ 14u + 1)
2
· (u
12
+ 6u
11
+ ··· + 848u + 64)
c
2
(u
4
− 2u
2
+ u + 1)
2
(u
7
+ u
6
− 3u
5
− 5u
4
+ u
3
+ 4u
2
+ u − 1)
· (u
10
− 3u
9
+ 4u
8
− 2u
7
+ 2u
6
− 4u
5
− u
3
+ 5u
2
− 2u − 1)
2
· (u
12
+ 8u
11
+ ··· + 52u + 8)
c
3
, c
10
(u
7
− 4u
5
+ 6u
3
− u
2
− 4u + 1)(u
8
− 4u
6
+ u
5
+ 5u
4
− u
3
− 4u
2
+ 1)
· (u
12
− 4u
10
+ 13u
8
+ 2u
7
− 24u
6
+ 3u
5
+ 18u
4
− 3u
3
− 7u
2
+ u + 1)
· (u
20
− u
19
+ ··· − 18u − 9)
c
4
, c
7
(u
7
− 4u
5
+ 6u
3
+ u
2
− 4u − 1)(u
8
− 4u
6
− u
5
+ 5u
4
+ u
3
− 4u
2
+ 1)
· (u
12
− 4u
10
+ 13u
8
+ 2u
7
− 24u
6
+ 3u
5
+ 18u
4
− 3u
3
− 7u
2
+ u + 1)
· (u
20
− u
19
+ ··· − 18u − 9)
c
5
(u
4
− 2u
2
− u + 1)
2
(u
7
− u
6
− 3u
5
+ 5u
4
+ u
3
− 4u
2
+ u + 1)
· (u
10
− 3u
9
+ 4u
8
− 2u
7
+ 2u
6
− 4u
5
− u
3
+ 5u
2
− 2u − 1)
2
· (u
12
+ 8u
11
+ ··· + 52u + 8)
c
6
, c
9
(u
7
+ u
6
+ u
4
− u
3
− u
2
− u − 1)(u
8
− 4u
7
+ ··· − 2u
2
+ 1)
· (u
12
+ u
11
+ ··· + 8u + 1)(u
20
− 3u
19
+ ··· − 6u − 1)
c
8
, c
11
(u
7
+ 8u
6
+ 28u
5
+ 56u
4
+ 68u
3
+ 49u
2
+ 18u + 1)
· (u
8
+ 8u
7
+ 26u
6
+ 49u
5
+ 61u
4
+ 49u
3
+ 26u
2
+ 8u + 1)
· (u
12
+ 8u
11
+ ··· + 15u + 1)(u
20
+ 5u
19
+ ··· + 576u + 81)
c
12
(u
4
− 4u
3
+ 4u
2
− u + 1)
2
· (u
7
+ 7u
6
+ 17u
5
+ 30u
4
+ 45u
3
+ 27u
2
+ 6u + 4)
· (u
10
− 4u
9
− u
8
+ 23u
7
− 31u
6
+ 7u
5
+ 6u
4
+ 4u
3
− 10u
2
+ 5u − 1)
2
· (u
12
+ 10u
11
+ ··· − 196u − 20)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
4
− 4y
3
− 2y
2
− 13y + 1)
2
· (y
7
− 7y
6
− 3y
5
− 133y
4
− 43y
3
+ 16y
2
+ 33y −1)
· (y
10
+ 15y
9
+ ··· − 154y + 1)
2
· (y
12
+ 18y
11
+ ··· − 419072y + 4096)
c
2
, c
5
(y
4
− 4y
3
+ 6y
2
− 5y + 1)
2
· (y
7
− 7y
6
+ 21y
5
− 37y
4
+ 37y
3
− 24y
2
+ 9y −1)
· (y
10
− y
9
+ 8y
8
− 12y
7
− 8y
6
+ 6y
5
− 4y
4
− 21y
3
+ 21y
2
− 14y + 1)
2
· (y
12
− 6y
11
+ ··· − 848y + 64)
c
3
, c
4
, c
7
c
10
(y
7
− 8y
6
+ 28y
5
− 56y
4
+ 68y
3
− 49y
2
+ 18y −1)
· (y
8
− 8y
7
+ 26y
6
− 49y
5
+ 61y
4
− 49y
3
+ 26y
2
− 8y + 1)
· (y
12
− 8y
11
+ ··· − 15y + 1)(y
20
− 5y
19
+ ··· − 576y + 81)
c
6
, c
9
(y
7
− y
6
− 4y
5
− y
4
+ 5y
3
+ 3y
2
− y −1)
· (y
8
− 6y
7
+ 9y
6
− 5y
5
+ 8y
4
+ y
3
− 4y
2
− 4y + 1)
· (y
12
+ 11y
11
+ ··· − 16y + 1)(y
20
+ 9y
19
+ ··· + 144y + 1)
c
8
, c
11
(y
7
− 8y
6
+ 24y
5
− 76y
4
+ 128y
3
− 65y
2
+ 226y −1)
· (y
8
− 12y
7
+ 14y
6
+ 39y
5
+ 145y
4
+ 39y
3
+ 14y
2
− 12y + 1)
· (y
12
+ 20y
11
+ ··· − 43y + 1)(y
20
+ 7y
19
+ ··· − 23004y + 6561)
c
12
(y
4
− 8y
3
+ 10y
2
+ 7y + 1)
2
· (y
7
− 15y
6
− 41y
5
+ 264y
4
+ 553y
3
− 429y
2
− 180y −16)
· ((y
10
− 18y
9
+ ··· − 5y + 1)
2
)(y
12
− 40y
11
+ ··· − 4496y + 400)
21
